$(20$ points$)$ A worker has utility function given $byU=u(c_1,c_2)$ where $c_{1}is$ her consumption

while she $is$ working and and $c_2$ her consumption after she retires. She earns zero income after

she retires $so$ she must save while she works for her consumption after her retirement. How will

she change $c_1,c_2$ and the amount $of$ saving $if$ the federal reserve raise the interest rate? Discuss

the income and substitution effects.

$(30$ points$)$ Consider the utility $ofan$ owner $of$ a convenient store $is$ given $byU=ln(w),$

where $wis$ his income. $He$ believes that the economy will $go$ into a recession $in$ the next year

with probability $of50\%.$ Let $w_R$ represent the owner's income $in$ the state $of$ recession and $w_{NR}$

the income when a recession does not occur, his expected utility will $be$ given $by$:

expected utility $=\frac{1}{2}ln(w_R)+\frac{1}{2}ln(w_{NR})$

$a$$$28,000in$ the state $ofno$

recession and $$10,000in$ the state $of$ recession, $or$ a necessity, which can bring $in$ $$19,000in$ the

state $ofno$ recession and $$15,000in$ the state $of$ recession. Which good will $he$ carry $in$ his store?

$(b)$ Suppose the store can use half $of$ its shelf space for each good. Would $he$ choose $todoso?$

Explain your answer.

$(c)$ What mix $of$ the luxury good and necessity would provide maximum expected utility for the

store owner?